A336226 Values z of primitive solutions (x, y, z) to the Diophantine equation x^3 + y^3 + 2*z^3 = 1458.

1, -3, 4, 9, -10, -12, 16, 21, 25, 37, -47, -48, 49, 64, -75, -87, 88, 100, 105, 121, 134, -147, 169, 172, -192, 196, -241, -243, 256, 289, -300, 361, -363, 400, 443, 484, -507, 529, 541, -588, 625, 676, -699, 732, -759, -768, 777, 784, 841, -867, 897, 961

Offset

1,2

Comments

Terms are arranged in order of increasing absolute value (if equal, the negative number comes first).

(11 + 3*n - 9*n^2)^3 + (11 + 3*(n + 1) - 9*(n + 1)^2)^3 + 2*(3*n + 1)^6 = 1458, the numbers of the form (3*n + 1)^2 are terms of the sequence.

(11 - 3*n - 9*n^2)^3 + (11 - 3*(n + 1) - 9*(n + 1)^2)^3 + 2*(3*n + 2)^6 = 1458, the numbers of the form (3*n + 2)^2 are also terms of the sequence.

Thus, A001651(n)^2 are terms of the sequence. There is an infinity of nontrivial solutions to the equation.

References

R. K. Guy, Unsolved Problems in Number Theory, D5.

Links

Table of n, a(n) for n=1..52.

Example

5^3 + 11^3 + 2 * 1^3 = 1458, 1 is a term.
(-1)^3 + (11)^3 + 2 * (4)^3 = 1458, 4 is a term.

Mathematica

Clear[t]
t = {};
Do[y = (1458 - x^3 - 2 z^3)^(1/3) /. (-1)^(1/3) -> -1; If[IntegerQ[y] && GCD[x, y, z] == 1, AppendTo[t, z]], {z, -980, 980}, {x, -25319, 25319}]
u = Union@t;
v = Table[(-1)^n*Floor[(n + 1)/2], {n, 0, 2000}];
Select[v, MemberQ[u, #] &]

Crossrefs

Cf. A000290, A000578, A001651, A003215, A004825, A004826, A050791, A130472, A195006.

Sequence in context: A275893 A325196 A242661 * A339658 A344297 A344292

Adjacent sequences: A336223 A336224 A336225 * A336227 A336228 A336229

Keyword

sign

Author

XU Pingya, Jul 17 2020

Status

approved